Let $X$ be a smooth complex projective variety, $\dim(X)>3$ and $D$ be a very ample possibly reducible divisor on $X$. Is it true that $\textrm{Pic}(X)\cong \textrm{Pic}(D)$? If not, what would be a reasonable condition on $D$ (for example, would it suffice that $D$ is irreducible)?

2 Answers
2

The answer is yes under very mild assumptions. In fact there is the following result.

Proposition. Let $L$ be a $k$-ample line bundle on a normal, irreducible, projective variety $X$ with at most Cohen-Macauley singularities. Assume that the dimension of the locus of non-rational singularities of $X$ is at most $0$ and that $D \in |L|$ is a divisor such that $X-D$ is a local complete intersection. Then, under the restriction mapping, we have $\textrm{Pic}(X) \cong \textrm{Pic}(D)$ if $\dim X \geq 4+k$ and $\textrm{Pic}(X) \to \textrm{Pic}(D)$ is injective with torsion free cokernel if $\dim X = 3+k$.

The case you are interested in corresponds to $k=0$. For instance, if $X$ is smooth (of dimension at least $4$) then all the assumptions are fulfilled and what you want is true.

The Proposition above is a consequence of Hamm's Lefschetz Theorem. For further details, see Beltrametti-Sommese, The adjunction theory of complex projective varieties, Corollary 2.3.4 page 51.

Remark. Given an integer $k \geq 0$, a line bundle $L$ on a projective variety $X$ is said to be $k$-ample if $a L$ is spanned for some $a > 0$, and the morphism $X \to \mathbb{P}^{h^0(aL)-1}$ defined by $|a L|$ has all fibers of dimension $\leq k$. For $k=0$, this is one of the basic characterization of ampleness.

Francesco Polizzi's answer is perfectly correct; also there are generalizations in the second part of Stratified Morse Theory by Goresky-MacPherson. However I want to point out that there is a different, much more algebraic perspective that works over an arbitrary field developed in SGA 2. Cohomologie Locale des Faisceaux Coherents et Theoremes de Lefschetz Locaux et Globaux, A. Grothendieck with an expose by Mme. M. Raynaud. The relevant result is Corollaire 3.6, Exp. XII, p. 153.